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Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies the 7 Properties for intersection numbers (I am not sure if these are well known or not, but I mean the ones from William Fultons book about algebraic curves).
My question is regarding the intersection number of two projective curves $F,G$, which we defined by $dim_k (\mathcal{O}_P (\mathbb{P}^2)/(F_*,G_*))$, where $F_*$ and $G_*$ are the corresponding dehomogenized forms. Our professor only told us that this satisfies the 7 properties again (however you have to change 2 of them a little so that they make sense in the projective case). Now I wanted to know if you actually have to check every single one of the properties again, or if there is a short argument tracing back to the affine case.
Thx in advance!

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    $\begingroup$ Since the definition only talks about local rings at a point, one may simply work in the affine open set containing that particular point. Thus we are reduced to the affine case. $\endgroup$ – random123 Dec 7 '18 at 8:09
  • $\begingroup$ @random123 I get it now, thank you! So does the uniqueness also follow like in the affine case? $\endgroup$ – user9620780 Dec 9 '18 at 20:12
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    $\begingroup$ I dont see why not. The same argument as above should apply here too. It should follow as long as its local in nature. That is depends only on the local rings. $\endgroup$ – random123 Dec 10 '18 at 3:32

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